Put it in the same directory as the SWF. Then you can give it a test
run in the AIR runtime with this command:

adl Connect4-app.xml

If everything is well-and-good, you'll get your app up in a native
window! There you go!

Technically, to build an AIR app, you should use amxmlc
instead of plain ol' mxmlc because it points to the AIR libs
that give you access to all the system-level functionality you need for
a real app. (Like making network connections, reading and writing files,
and exiting the app, for example.)

Now, actually building an AIR application to be distributed is a bit
of a process that I haven't explored. It involves digitally
signing the package; you can get a cert from
a CA,
or just self-sign
your own (though some users might not trust self-signed apps.) The
reasoning behind the signature is to provide a level of accountability
(you know who wrote the app) and security (you know the app hasn't been
tampered with) for the application. This is important, since the app has
access to your network and local data, just like any other native
application. The signing process doesn't look too bad to go
through, and it can be automated in a Makefile once you have
your cert.

But I'm giving up for the moment—I'll revisit it later. Adobe really
should work hard on their well-deserved 64-bit reputation.

If you're running Windows or OS X, or 32-bit Linux with KDE or GNOME,
it should be pretty seamless. Just pop over to the Adobe
AIR marketplace and grab some free apps. It'll auto-install the
runtime if you don't have it already.